Zeev DvirZeev DvirWeizmann Institute of Weizmann Institute of

ScienceScience Amir ShpilkaAmir Shpilka

TechnionTechnion

Locally decodable codes with 2 Locally decodable codes with 2 queriesqueries

andand polynomial identity testing for depth polynomial identity testing for depth

3 circuits3 circuits

This talkThis talk• Explaining the title:Explaining the title:

– Locally Decodable codesLocally Decodable codes– Polynomial identity testingPolynomial identity testing– depth 3 circuitsdepth 3 circuits

• Results:Results:– Improved bounds for 2-queries LDC'sImproved bounds for 2-queries LDC's– Getting 2-LDC's from identically zero Getting 2-LDC's from identically zero

depth 3 circuits.depth 3 circuits.– Structural theorem for identically zero Structural theorem for identically zero

depth 3 circuitsdepth 3 circuits– PIT for depth 3 circuits PIT for depth 3 circuits

Locally decodable codesLocally decodable codes

xx11 ......

xxkk ...... xxnn yy11 yy22 ...... yyii ......

yyjj ...... yymm

xxkk

Def:Def: E: E: FFnn !! FFmm is is q-LDCq-LDC if x if xkk can be can be recovered from q entries of E(x). recovered from q entries of E(x).

Even if E(x) is corrupted in Even if E(x) is corrupted in m coordinates. m coordinates.

With high probability.With high probability.

AlgorithmAlgorithmk

w.h.p

Main questionsMain questions: constructing LDC's, : constructing LDC's, proving lower bounds on their length.proving lower bounds on their length.

Known constructions: Known constructions: q-LDC E: q-LDC E: FFnn !! FFmm with with m = exp(nm = exp(nloglog(q)/qloglog(q)/q¢¢log(q)log(q)) [) [BIKR02BIKR02].].

Lower boundsLower bounds::[[KT00KT00]: m = ]: m = (n(n1 + 1/q-11 + 1/q-1))[[GKST01GKST01]: In ]: In linear linear 2-LDC over 2-LDC over FF

m = exp(m = exp((n)- log|(n)- log|FF|)|)[[KdW03KdW03]: In 2-LDC over {]: In 2-LDC over {0,10,1} } m = m =

exp(exp((n)).(n)).

Our resultOur result: In : In linearlinear 2-LDC 2-LDC m = exp(m = exp((n)).(n)). Works for every field size, i.e. Works for every field size, i.e. FF==RR..

AssumptionAssumption: f has succinct representation.: f has succinct representation.

Motivation: Motivation: Natural problem, many Natural problem, many applications: primality testing, finding applications: primality testing, finding matching ...matching ...

Schwartz-ZippelSchwartz-Zippel: Evaluate f(x) at a random : Evaluate f(x) at a random point.point.

Long Term GoalsLong Term Goals: Deterministic algorithm.: Deterministic algorithm.

Short Term GoalsShort Term Goals: Restricted Models.: Restricted Models.

Polynomial identity testingPolynomial identity testing

f(x1,...,xn) 0 ?

General circuits: General circuits: Randomized algorithms Randomized algorithms [S80],[Z79],[CK97],[LV98],[AB03][S80],[Z79],[CK97],[LV98],[AB03]::

poly(poly(11//,size) time, n,size) time, n¢¢log(d/log(d/) random bits) random bits

Hardness vs. Randomness trade-off:Hardness vs. Randomness trade-off: [[KI03KI03] ]

PIT PIT 22 P P )) arithmetic lower bound for NEXP arithmetic lower bound for NEXP– NEXP NEXP ** P/poly P/poly oror – PERM PERM arithmetic P/poly arithmetic P/poly

Lower bounds for arithmetic circuits imply Lower bounds for arithmetic circuits imply sub-exponential time deterministic algs.sub-exponential time deterministic algs.

look for PIT where l.b. are known!look for PIT where l.b. are known!

Non-commutative formulasNon-commutative formulas: (vars do not commute): (vars do not commute)[N91][N91] exponential lower bound on formula size exponential lower bound on formula size[RS04] [RS04] PIT determ. poly-time in size of formulaPIT determ. poly-time in size of formula

““Depth 2” circuits: Depth 2” circuits: (sparse polynomials)(sparse polynomials)[BoT88],[GKS90],...,[KS01][BoT88],[GKS90],...,[KS01]: deterministic poly time.: deterministic poly time.

No sub-exp time deterministic algs. for depth > 2No sub-exp time deterministic algs. for depth > 2OpenOpen [ [KS01KS01]: depth 3 circuits w. top fan-in = 3.]: depth 3 circuits w. top fan-in = 3.

This paperThis paper: depth 3 circuits with small top fan-in:: depth 3 circuits with small top fan-in:deterministicdeterministic: quasi-polynomial time PIT alg. : quasi-polynomial time PIT alg.

(poly time for (poly time for multilinearmultilinear circuits). circuits).randomizedrandomized: polynomial time polylog random : polynomial time polylog random bits.bits.

New resultNew result: [: [KS06KS06] polynomial time algorithm.] polynomial time algorithm.

Depth 3 circuits - Depth 3 circuits - (k) circuits(k) circuits

+

XX XX XX XX

+++

top fan-in

x1 xn

ckc1

a1 an

1

a0

Li,j = t=1...n at¢xt + a0

Mi = j=1...diLi,j

...

M1

MkL1,1

C(x) = i=1...k ci¢Mi = ici

¢jLi,j

L1,d1

What's next:What's next:• Sketch of lower bound for 2-LDC.Sketch of lower bound for 2-LDC.• PIT for depth 3 circuits with top fan-in = 2.PIT for depth 3 circuits with top fan-in = 2.• PIT for depth 3 circuits with top fan-in = 3.PIT for depth 3 circuits with top fan-in = 3.• General depth 3 circuits (sketch)General depth 3 circuits (sketch)• Structural theorem for identically zero Structural theorem for identically zero

depth 3 circuits.depth 3 circuits.• PIT algorithms for depth 3 circuitsPIT algorithms for depth 3 circuits

Thm 1Thm 1 [ [GKST01GKST01]: For any linear 2-LDC over {0,1} of ]: For any linear 2-LDC over {0,1} of length m, m = exp(length m, m = exp((n)).(n)).

ProofProof:: Isoperimetric inequality.Isoperimetric inequality.

Thm 2Thm 2 [ [GKST01GKST01]: For any linear 2-LDC over ]: For any linear 2-LDC over FF of of length m, m = exp(length m, m = exp((n) – log|(n) – log|FF|).|).

Proof: Proof: combine next lemma with theorem 1.combine next lemma with theorem 1.

LemmaLemma [ [GKST01GKST01]: If ]: If 99 linear 2-LDC over linear 2-LDC over F F of length of length m then m then 99 linear 2-LDC over {0,1} of length | linear 2-LDC over {0,1} of length |FF||¢¢ m. m.

ProofProof: : randomly map all multiples of all coordinates to randomly map all multiples of all coordinates to {0,1}.{0,1}.

New LemmaNew Lemma: If : If 99 linear 2-LDC over linear 2-LDC over F F of length m of length m then then 99 linear 2-LDC over {0,1} of length m. linear 2-LDC over {0,1} of length m.

ProofProof: : randomly map randomly map well chosen multiplewell chosen multiple of each of each coordinate to {0,1}.coordinate to {0,1}.

What's next:What's next: Sketch of lower bound for 2-LDC.Sketch of lower bound for 2-LDC.• PIT for depth 3 circuits with top fan-in = 2.PIT for depth 3 circuits with top fan-in = 2.• PIT for depth 3 circuits with top fan-in = 3.PIT for depth 3 circuits with top fan-in = 3.• General depth 3 circuits (sketch)General depth 3 circuits (sketch)• Structural theorem for identically zero Structural theorem for identically zero

depth 3 circuits.depth 3 circuits.• PIT algorithms for depth 3 circuitsPIT algorithms for depth 3 circuits

Identically zero Identically zero (2) circuits(2) circuits

ReminderReminder: C(x) = c: C(x) = c11¢¢MM11 + c + c22¢¢MM22

LL11 LL22 ...... LLii ......

LLjj ...... LLdd

L'L'11 L'L'22 ...... L'L'ii ......

L'L'jj ...... L'L'dd

M1(x)=

M2(x)=FactFact: linear functions are irreducible : linear functions are irreducible polynomial.polynomial.

CorollaryCorollary: C: C0 then M0 then M11, M, M22 have the same have the same factors.factors.

CorollaryCorollary: : 99 matching i matching i j(i) s.t. L j(i) s.t. Lii ~ L' ~ L'j(i)j(i)

PIT algorithmPIT algorithm: look for such a matching.: look for such a matching.

What's next:What's next: Sketch of lower bound for 2-LDC.Sketch of lower bound for 2-LDC. PIT for depth 3 circuits with top fan-in = 2.PIT for depth 3 circuits with top fan-in = 2.• PIT for depth 3 circuits with top fan-in = 3.PIT for depth 3 circuits with top fan-in = 3.• General depth 3 circuits (sketch)General depth 3 circuits (sketch)• Structural theorem for identically zero Structural theorem for identically zero

depth 3 circuits.depth 3 circuits.• PIT algorithms for depth 3 circuitsPIT algorithms for depth 3 circuits

PreliminariesPreliminariesClaimClaim: wlog linear functions are : wlog linear functions are

homogeneous (no constant term).homogeneous (no constant term).

ClaimClaim: A:: A:FFnnFFnn invertible linear map, then invertible linear map, then C(x)C(x)0 0 , , C(AC(A¢¢x)x)0.0.

DefinitionDefinition:: rr ,, rank(C) rank(C) ,, rank(linear functions in C). rank(linear functions in C).

Corollary 1Corollary 1: wlog L: wlog Lii's depend only on 's depend only on xx11,...,x,...,xrr..

Corollary 2Corollary 2: wlog x: wlog x11,...,x,...,xrr appear as linear appear as linear functions in C.functions in C.

InputInput: C(x) = c: C(x) = c11¢¢MM11 + c + c22¢¢MM2 2 + c+ c33¢¢MM33

xx11,...,x,...,xrr are linear functions in C are linear functions in C

wlog assumewlog assume g.c.d.(M g.c.d.(M11,M,M22,M,M33) = 1) = 1

M2(x)=

LL2d+12d+1 LL2d+22d+2 ...... LL2d+i2d+i ...... xxtt ...... LL3d3dM3(x)=

M1(x)=

Idea: Idea: reduction to reduction to (2): C(2): C0 0 ) ) C|C|xxss=0 =0 0 0

) ) if xif xss22MM11 then c then c22¢¢MM22||xxss=0=0+c+c33¢¢MM33||xxss=0=0=0.=0.

LemmaLemma: : 88xxss 99d pairs (d pairs (i,j(i)i,j(i)) s.t. L) s.t. Lii||xxss=0 =0 ~ L~ Lj(i)j(i)||xxss=0=0

0

LLd+1d+1 LLd+2d+2 ...... LLd+id+i ...... LLd+jd+j ...... LL2d2d

LL11 LL22 ...... LLii ...... xxss ...... LLdd

0

LemmaLemma: i=: i=1,21,2 L Lii22MMii: L: L11||xxss=0 =0 ~ L~ L22||xxss=0=0 ) ) xxss 22

span(Lspan(L11,L,L22))

ProofProof: Otherwise L: Otherwise L1 1 ~ L~ L22 )) L L11 | M | M11,M,M22

)) if C if C0 then L0 then L11 | M | M33 ) ) LL11 22 g.c.d(M g.c.d(M11,M,M22,M,M33) ) ??

DefineDefine E(x) = L E(x) = L11(x),...,L(x),...,L3d3d(x)(x)

ClaimClaim: : 88s s 99d pairs (i,j(i)) s.t. xd pairs (i,j(i)) s.t. xss 22 span(E(x) span(E(x)ii,E(x),E(x)j(i)j(i)).).

CorollaryCorollary: E is a 2-LDC of length 3d.: E is a 2-LDC of length 3d.

CorollaryCorollary: 3d=exp(: 3d=exp((r)) (r)) ) ) r=O(log(d))r=O(log(d))..

ThmThm: : If CIf C0 is 0 is (3) then rank(C) = O(log(d)).(3) then rank(C) = O(log(d)).

PIT AlgorithmPIT Algorithm: brute force. time = exp(log(d): brute force. time = exp(log(d)22).).

InputInput: C(x) = c: C(x) = c11¢¢MM11 + c + c22¢¢MM2 2 + c+ c33¢¢MM33

MM11 = = i=1...di=1...dLLii(x), M(x), M22==i=1...di=1...dLLd+id+i(x), M(x), M33==i=1...di=1...dLL2d+i2d+i(x) (x)

xx11,...,x,...,xrr are linear functions in C are linear functions in C

LemmaLemma: : 88xxss 99d pairs (d pairs (i,j(i)i,j(i)) s.t. L) s.t. Lii||xxss=0 =0 ~ L~ Lj(i)j(i)||xxss=0=0

What's next:What's next: Sketch of lower bound for 2-LDC.Sketch of lower bound for 2-LDC. PIT for depth 3 circuits with top fan-in = 2.PIT for depth 3 circuits with top fan-in = 2. PIT for depth 3 circuits with top fan-in = 3.PIT for depth 3 circuits with top fan-in = 3.• General depth 3 circuits (sketch)General depth 3 circuits (sketch)• Structural theorem for identically zero Structural theorem for identically zero

depth 3 circuits.depth 3 circuits.• PIT algorithms for depth 3 circuitsPIT algorithms for depth 3 circuits

InputInput: C(x) = c: C(x) = c11¢¢MM11 + c + c22¢¢MM2 2 + c+ c33¢¢MM33 + ... + c + ... + ckk¢¢MMkk

DefDef: C is simple if g.c.d.(M: C is simple if g.c.d.(M11,...,M,...,Mkk)=1)=1

DefDef: : sim(C)sim(C) = C/g.c.d.(C) = C/g.c.d.(C)

DefDef: C is minimal if no sub-circuit is zero. : C is minimal if no sub-circuit is zero.

ThmThm: C: C0 is simple and minimal, r = rank(C), 0 is simple and minimal, r = rank(C), d = deg(C). Then d = deg(C). Then 99 2-LDC E: 2-LDC E: FFaa FFbb s.t. s.t.

aa = r/ = r/22kk22log(d)log(d)k-3k-3 b b = kd = kd

CorollaryCorollary: rank(C) : rank(C) ·· O O((log(d)log(d)k-2k-2))ProofProof: induction on k. Assume x: induction on k. Assume x11,...,x,...,xrr 22 C. C.

Consider C|Consider C|xxii = 0 = 0 0. Top fan-in is k-1. Done? 0. Top fan-in is k-1. Done?

simple? minimal? rank?simple? minimal? rank?

M2(x)=

LL3,13,1 LL3,23,2 ...... LL3,i3,i ...... xxtt ...... LL3,d3,dM3(x)=

M1(x)=

0

LL2,12,1 LL2,22,2 ...... LL2,i2,i ...... LL2,j2,j ...... LL2,d2,d

LL1,11,1 LL1,21,2 ...... LL1,i1,i ...... xxss ...... LL1,d1,d

LLk,1k,1 LLk,2k,2 ...... LLk,ik,i ...... LLk,jk,j ...... LLk,dk,dMk(x)=

Is

...

Claim: 8xs 9Is s.t. (CIs)|xs=0 0 and minimal

Cor: 9I,r' ¸ r/2k s.t. (wlog) 8 1· s · r' (CI)|xs=0 0 and minimal.

Cor: 9I,r' ¸ r/2k s.t. 8 1 · s · r' (CI)|xs=0 0 and minimal.

Optimistic: done?

Problematic: what's the rank of (CI)|xs=0 ?

Optimistic: lemma: rank(CI) ¸ r' ¸ r/2k

Problematic: (CI)|xs=0 not simple

Optimistic: consider sim((CI)|xs=0 ) (removing g.c.d.)

Problematic: what happens to the rank?

Optimistic: eh ...

Lemma: 9 xi s.t.

rank(sim((CI)|xs=0)) ¸ rank(CI)/2klog(d)

Proof: …

End of proof: induction on (CI)|xi=0 (from Lemma).

What's next:What's next: Sketch of lower bound for 2-LDC.Sketch of lower bound for 2-LDC. PIT for depth 3 circuits with top fan-in = 2.PIT for depth 3 circuits with top fan-in = 2. PIT for depth 3 circuits with top fan-in = 3.PIT for depth 3 circuits with top fan-in = 3. General depth 3 circuits (sketch)General depth 3 circuits (sketch)• Structural theorem for identically zero Structural theorem for identically zero

depth 3 circuits.depth 3 circuits.• PIT algorithms for depth 3 circuitsPIT algorithms for depth 3 circuits

Structural theoremStructural theorem: C : C 0 is 0 is (k) then:(k) then:

99 partition I partition I11 t t II22 t t ... ... tt I Imm = [k] s.t. = [k] s.t.

CCIjIj 0 minimal 0 minimal (C = C(C = CII11 + C + CII22

+ ... + C + ... + CIImm))

rank(sim(Crank(sim(CIjIj)) )) ·· O(log(d) O(log(d)|I|Ijj|-2|-2))

PIT algorithmPIT algorithm: For each I : For each I ½½ [k] check [k] check whether rank(sim(Cwhether rank(sim(CII)) )) ·· O(log(d) O(log(d)|I|-2|I|-2) )

if yes then brute force check if Cif yes then brute force check if CII 0 0

if if 99 partition as in theorem then C partition as in theorem then C 0 0

Running time: Running time: exp(log(d)exp(log(d)k-1k-1))..

The Multilinear CaseThe Multilinear Case

If C is multilinear then If C is multilinear then rank(C)=d.rank(C)=d.

But we proved that if C=0 is simple But we proved that if C=0 is simple and minimal then and minimal then rank(C) rank(C) ·· polylog(d) polylog(d)

We get that We get that d d · · polylog(d) polylog(d)

Can only hold for finitely many values !Can only hold for finitely many values !

Conclusion: Conclusion: d d ·· O(1) O(1)

rank(C) rank(C) ·· dk dk ·· O(1) O(1)

Polynomial time algorithmPolynomial time algorithm

Open problemsOpen problems

• PIT algorithms for stronger models:PIT algorithms for stronger models:– Depth 3 circuitsDepth 3 circuits– Bounded depth Bounded depth

• Tightness of our results:Tightness of our results:– ConjectureConjecture: If C : If C 0 is 0 is (k) simple, (k) simple,

minimal then rank(C) = poly(k).minimal then rank(C) = poly(k).– [[KS06KS06] ] Not true for finite fields! Not true for finite fields! Example in of a circuit with top Example in of a circuit with top

fanin=3 and rank ~ log(d)fanin=3 and rank ~ log(d)

((MultilinearMultilinear))(Multilinear)(Multilinear)